where \(X_t\)  represents the segment of the path of the process \(X(\cdot )\)  from \(t-\delta \)  to \(t\) . A special case is that \(X_t=X(t-\delta )\)  as in Chen–Wu [1]}, [2]}. The function \(f\)  can be reasonably specific. For example, \(f(X_t,\pi (t))=K f(X_t)^{\alpha } \pi ^{\beta }(t)\) , where \(K, \alpha , \beta \)  are some constants as in Gandolfo [3]}. For simplicity, similar to Ivanov–Swishchuk [4]}, we set \(\alpha =\beta =1\)  to rewrite our model as
\(\left\lbrace \begin{aligned}\displaystyle \mathrm {d} X(t)=&\ [K\pi (t) f(X_t)-c(t)] \mathrm {d} t+\sigma (X_t) \mathrm {d} W(t),\quad t\in [0,T],\\{\smallskip }\displaystyle X(t)=&\ \gamma (t), \quad t \in [-\delta , 0],\end{aligned}\right.\) 
